Do I need to keep my fractured wrist in a sling?

No. A sling will just make your elbow more stiff. No need to immobilize the elbow with a wrist injury.
NO. The type of immobilization depends on the type of wrist fracture. In some cases the elbow does need to be immobilized but a sling is rarely the answer. Best to have the wrist splinted and the hand and wrist elevated.